Zero Element Generates Null Ideal

Theorem

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

For $r \in R$, let $\ideal r$ denote the ideal generated by $r$.

Then $\ideal {0_R}$ is the null ideal.

Proof

By definition:

$\ideal {0_R} = \set {r \circ 0_R: r \in R}$

but for each $r \in R$ we have by Ring Product with Zero that $r \circ 0_R = 0_R$ for all $r \in R$.

Therefore $\ideal {0_R}$ is the null ideal.

$\blacksquare$